(6-3)   ①   The starting equation is:

The definitions we then need are: p  =  mv  and Ek  = ½ mv2.

Hence,  p2  =  m2v2  so Ek = p2/2m  and therefore:     p2  =  2mEk

Substituting this into the above equation, we get:

②  Substituting in  Ek  =  E – V(x),  we get:

③  This rearranges to give  :  

And this, at last, is the Schrödinger equation – the equation that defines the wavefunction associated with a particle.

So the Hamiltonian that operates on ψ(x) is: